The velocity potential and the interacting force for two spheres moving perpendicularly to the line joining their centers L. Li W. W. Schultz H. MerteJr. Article

Received: 25 September 1991 Accepted: 20 July 1992 DOI :
10.1007/BF00127479

Cite this article as: Li, L., Schultz, W.W. & Merte, H. J Eng Math (1993) 27: 147. doi:10.1007/BF00127479
Abstract The velocity potential around two spheres moving perpendicularly to the line joining their centers is given by a series of spherical harmonics. The appropriateness of the truncation is evaluated by determining the residual normal surface velocity on the spheres. In evaluating the residual normal velocity, a recursive procedure is constructed to evaluate the spherical harmonics to reduce computational effort and truncation error as compared to direct transformation or numerical integration. We estimate the lift force coefficient for touching spheres to be 0.577771, compared to the most accurate earlier estimate of 0.51435 by Miloh (1977).

References 1.

D. Leighton and A. Acrivos, The lift on a small sphere touching a plane in the presence of a simple shear flow. ZAMP 36 (1985) 174–178.

2.

D.A. Drew, The lift force on a small sphere in the presence of a wall. J. of Chemical Engineering Science 43 (4) (1988) 769–773.

3.

P. Cherukat and J.B. McLaughlin, Wall-induced lift on a sphere. Int. J. Multiphase Flow 16 (5) (1990) 899–907.

4.

P. Vasseur and R.G. Cox, The lateral migration of spherical particles sedimenting in a stagnant bounded fluid. J. Fluid Mech. 80 (1977) 561–591.

5.

W.M. Hicks, On the motion of two spheres in a fluid. Philosophical Transaction. Royal Society of London 171 (1880) 455–492.

6.

A.B. Basset, On the Motion of Two Spheres in a Liquid, and Allied Problems. Proceedings, Mathematical Society 18 (1887) 369–377.

7.

R.A. Herman, On the motion of two spheres in fluid and allied problems. Quartly Journal of Pure and Applied Mathematics xxii (1887) 204–262.

8.

D. Endo, The Force on Two Spheres Placed in Uniform Flow. Proc. Phys.-Math. Soc. Japan 20 (1938) 667–703.

9.

J.D. Love, Dielectric sphere-sphere and sphere-plane problems in electrostatics. Quartly Journal of Mech. Appl. Math. 28 (4) (1975) 449–471.

10.

R.D. Small and D. Weihs, J. of Applied Mechanics. Trans. A.S.M.E. 42 (1975) 763.

11.

O.V. Voinov, On the motion of two spheres in a perfect fluid. J. of Applied Mathematics and Mechanics 33 (4) (1969) 638–646.

12.

T. Miloh, Hydrodynamics of deformable contiguous spherical shapes in an incompressible inviscid fluid. J. of Engineering Mathematics 11 (4) (1977) 349–372.

13.

E.W. Hobson, Theory of spherical and ellipsoidal harmonics . Cambridge University Press (1931).

14.

R.L. Burden, and J.D. Faires, Numerical Analysis , 3rd edn. Prindle, Weber & Schmidt, 1985.

© Kluwer Academic Publishers 1993

Authors and Affiliations L. Li W. W. Schultz H. MerteJr. 1. Department of Mechanical Engineering University of Michigan Ann Arbor USA